Finite Group Databasebeta

← #28484732

Subgroups — #28484732

Order 15104.

684 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429901C211yesmaximal, characteristic
2#21422667C212yesmaximal, complemented
3C2×#16837126C211yesmaximal, characteristic
4#21421963C212yesmaximal, complemented
5C59×#2315C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7C2×C22⋊(C59⋊C8)C2212yes—
8#16837338C412yescomplemented
9#16837325—22no—
10#16837314C412yescomplemented
11#16837275—21no—
12#16837262C2211yescharacteristic
13#16837258—21no—
14#16837138—24no—
15#16837126—22no—
16C22×C22⋊C4×C59—21no—
17C22×C22⋊C4×C59C2211yescharacteristic
18C59×#385—24no—
19C59×#380—21no—
20C59×#380C2212yes—
21C22×Dic118C2×C411yescharacteristic, complemented
22C22×C59⋊C8—22no—
23#14273706—22no—
24#14273706—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59C2×C411yescharacteristic, complemented
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C22⋊(C59⋊C8)—22no—
34C22⋊(C59⋊C8)—22no—
35C59⋊#95—22no—
36C59⋊#95C2×C412yescomplemented
37C25×C59D411yescharacteristic
38C23×C4×C59—22no—
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—42no—
48C2×C22⋊C4×C59—42no—
49C2×C22⋊C4×C59C2×C412yes—
50C2×C22⋊C4×C59D412yes—
51#95×C59—22no—
52#95×C59D414yes—
53C24×C59—21no—
54C24×C59—21no—
55C24×C59—21no—
56C24×C59—21no—
57C24×C59—81no—
58C24×C59—81no—
59C24×C59C22⋊C412yes—
60C24×C59SD1612yes—
61C24×C59Q1612yes—
62C24×C59C2×D411yescharacteristic
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—24no—
66C22×C4×C59—44no—
67C22×C4×C59—81no—
68C22×C4×C59—81no—
69C22×C4×C59—82no—
70C22×C4×C59—82no—
71C22×C4×C59C22⋊C412yes—
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D412yes—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59#4112yes—
76C22×Dic59—21no—
77C22×Dic59—22no—
78C22×Dic59—21no—
79C2×Dic118—22no—
80C2×Dic118—81no—
81C4⋊C4×C59—82no—
82C4⋊C4×C59—82no—
83C22⋊C4×C59—24no—
84C22⋊C4×C59—24no—
85C22⋊C4×C59—44no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—42no—
88C22⋊Dic59—41no—
89C22⋊Dic59—41no—
90C22⋊Dic59—41no—
91C22⋊Dic59—41no—
92C22⋊Dic59—82no—
93C22⋊Dic59—82no—
94C4⋊Dic59—22no—
95C4⋊Dic59—81no—
96Dic59⋊C4—42no—
97Dic59⋊C4—82no—
98C2×C59⋊C8—42no—
99C2×C59⋊C8—82no—
100C23×C59—22no—
101C23×C59—22no—
102C23×C59—22no—
103C23×C59—22no—
104C23×C59—22no—
105C23×C59—22no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—42no—
109C23×C59—42no—
110C23×C59—42no—
111C23×C59—42no—
112C23×C59—81no—
113C23×C59—81no—
114C23×C59—82no—
115C23×C59—82no—
116C23×C59—82no—
117C23×C59—82no—
118C23×C59#10314yes—
119C23×C59C2×C22⋊C411yescharacteristic
120C23×C59#12011yescharacteristic
121C23×C59#12311yescharacteristic
122C23×C59C2×SD1611yescharacteristic
123C23×C59C2×Q1611yescharacteristic
124C23×C59AGL(1,ℤ/8)11yescharacteristic
125C23×C59#13711yescharacteristic
126C2×C4×C59—22no—
127C2×C4×C59—42no—
128C2×C4×C59—41no—
129C2×C4×C59—42no—
130C2×C4×C59—41no—
131C2×C4×C59—42no—
132C2×C4×C59—42no—
133C2×C4×C59—44no—
134C2×C4×C59—44no—
135C2×C4×C59—44no—
136C2×C4×C59—81no—
137C2×C4×C59—88no—
138C2×C4×C59C4×D412yes—
139C2×C4×C59#12112yes—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59—42no—
143C2×Dic59—42no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—42no—
147Dic118—82no—
148C59⋊C8—82no—
149#9199—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—24no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—44no—
162C22×C59—44no—
163C22×C59—44no—
164C22×C59—42no—
165C22×C59—42no—
166C22×C59—82no—
167C22×C59—82no—
168C22×C59—82no—
169C22×C59#38611yescharacteristic
170C22×C59C2×#10311yescharacteristic
171C22×C59#41711yescharacteristic
172C22×C59#45011yescharacteristic
173C22×C59#45111yescharacteristic
174C22×C59#48311yescharacteristic
175C22×C59#48411yescharacteristic
176C4×C59—22no—
177C4×C59—22no—
178C4×C59—42no—
179C4×C59—84no—
180Dic59—82no—
181Dic59—81no—
182Dic59—81no—
183#2626—591no—
184#2315D5911yescharacteristic
185#1520—592no—
186C2×#323—591no—
187#1289—592no—
188C2×C59—21no—
189C2×C59—21no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—44no—
195C2×C59#187211yescharacteristic
196C2×C59#191111yescharacteristic
197C2×C59#191311yescharacteristic
198C2×C22⋊Q8—591no—
199C22×C22⋊C4—21no—
200C22×C22⋊C4C2×D5911yescharacteristic
201C2×C22⋊C8—592no—
202#399—592no—
203#396—1182no—
204#393—592no—
205#385—24no—
206#385—1181no—
207#380—21no—
208#380—591no—
209#380C2×D5912yes—
210#379—1181no—
211#328—1184no—
212#323—1182no—
213C59#919911yescharacteristic, complemented, Sylow
214C25C22⋊D5911yescharacteristic
215C22×Q8—591no—
216C23×C4—22no—
217C23×C4C22⋊D5911yescharacteristic
218C23×C4C22×D5911yescharacteristic
219C22×C8—1182no—
220C22⋊Q8—1182no—
221C22⋊Q8—1182no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—591no—
225C2×C4⋊C4—1182no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4—1181no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—1182no—
235C2×C22⋊C4—1182no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1181no—
238C2×C22⋊C4C4×D5912yes—
239C2×C22⋊C4D23612yes—
240C22⋊C8—1182no—
241C22⋊C8—1182no—
242#95—22no—
243#95—592no—
244#95—1182no—
245#95C22⋊D5914yes—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24(C2×D59)⋊C412yes—
253C24#1174567012yes—
254C24#1174567212yes—
255C24C2×C22⋊D5911yescharacteristic
256C2×Q8—1182no—
257C2×Q8—4721no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—44no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—1182no—
267C22×C4—1181no—
268C22×C4—1181no—
269C22×C4(C2×D59)⋊C412yes—
270C22×C4C2×C4×D5911yescharacteristic
271C22×C4C2×D23611yescharacteristic
272C22×C4#1174568512yes—
273C22×C4C2×C22⋊D5912yes—
274C2×C8—4722no—
275C2×C8—2362no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—1182no—
279C4⋊C4—4722no—
280C4⋊C4—4721no—
281C4⋊C4—2362no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—4722no—
288C22⋊C4—4722no—
289C22⋊C4—2361no—
290C22⋊C4—2361no—
291C22⋊C4—2361no—
292C22⋊C4—2361no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1427365014yes—
312C23C2×(C2×D59)⋊C411yescharacteristic
313C23#1427369611yescharacteristic
314C23#1427371011yescharacteristic
315C23C2×#1174567011yescharacteristic
316C23C2×#1174567211yescharacteristic
317C23#1427375311yescharacteristic
318C23#1427375811yescharacteristic
319Q8—4722no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—2362no—
333C2×C4—2362no—
334C2×C4—2362no—
335C2×C4—2362no—
336C2×C4—2362no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4C4×C22⋊D5912yes—
340C2×C4#1427370112yes—
341C8—4722no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1683728111yescharacteristic
362C22C2×#1427365011yescharacteristic
363C22#1683740111yescharacteristic
364C22#1683755311yescharacteristic
365C22#1683755811yescharacteristic
366C22#1683773211yescharacteristic
367C22#1683773711yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—84no—
372C4—4722no—
373C4—4721no—
374C4—4721no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2142440311yescharacteristic
383C2#2142462811yescharacteristic
384C2#2142463811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2315C2
socleC22×C59#386
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#2315D59
hypocenterC59#9199
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21422667C2semidirect
4. #21421963C2semidirect
8. #16837338C4semidirect
10. #16837314C4semidirect
21. C22×Dic118C2×C4semidirect
26. C2×C4⋊Dic59C2×C4semidirect
36. C59⋊#95C2×C4semidirect
213. C59#9199semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214299012
#214226672
C2×#168371262
#214219632
C59×#23152
#919959